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Find the remainder when 5,142,376,298 is divided by 9?
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The long division method will definitely tell you the remainder of dividing 5,142,376,298 by 9 but you can very well imagine how long would that take.
The good news is that just like the divisibility rules, there are the remainder rules.
These can help you quickly find the remainder without doing the long division. Let’s confine these rules to dividing by the following numbers:
2, 3, 4, 5, 8, 9, 10
Note that the remainder rules of 6 and 7 are omitted as I will discuss them in a separate post.
Remainder Rule when dividing by 2 – well, that’s easy. If an integer is divisible by 2 then the remainder is 0, otherwise, the remainder is 1.
For example, 1,456 is divisible by 2, so the remainder is 0.
Similarly, 2,390,399 is not divisible by 2, hence the remainder is 1.
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What is the remainder when -15,765 is divided by 2? Since ?15,765 is not divisible by 2, therefore the remainder is 1. |
Remainder Rule when dividing by 3 – An integer and the sum of its digits both leave the same remainder on division by 3.
Therefore, to find the remainder when an integer is divided by 3, first find the sum of its digits. Then divide the sum by 3 to find the remainder.
For example, take 38. The sum of its digits is 3 + 8 = 11. Now divide the sum (11) by 3 to find the remainder, which is 2.
Using long division, when we divide 38 by 3, we get the same remainder, which is 2.
This method can be extremely helpful with large numbers.
Let’s take another example; 34,259,677,858.
The sum of the digits is 3+4+2+5+9+6+7+7+8+5+8 = 64. When we divide 64 by 3, we get a remainder of 1.
- Note that you could also have again taken the sum of the digits of 64, 6+4=10, and then divided 10 by 3 to get the remainder, which is 1.
| What is the remainder when 13,455,765 is divided by 3? |
The sum of the digits of 13,455,765 is:
1+3+4+5+5+7+6+5 = 36. Now, the sum of the digits of 36 is 3+6 = 9. When we divide 9 by 3, we get a remainder of 0, i.e. 13,455,765 is divisible by 3.
This is the same remainder that you would get if you do the long division on 13,455,765 ÷ 3.
Remainder Rule when dividing by 4 – first form a two-digit number from the last two digits of the integer. Then divide the two-digit number by 4 and find the remainder.
For example, take 12,567. Its last two digits form a two-digit number, 67. Then divide 67 by 4 using the long division. You get a remainder of 3.
| What is the remainder when 5,478,953,458 is divided by 4? |
The last two digits of 5,478,953,458 form a two-digit number, 58. Then divide 58 by 4 using the long division. You get a remainder of 2.
Remainder Rule when dividing by 5.
To find the remainder when dividing a number by 5, simply divide the last digit (the unit’s digit) by 5 to find the remainder.
For example, take 3,569. The last digit (unit’s digit) is 9. Divide 9 by 5 to find the remainder, which is 4.
Let’s do another example. Take 943,840. The last digit (unit’s digit) is 0. Divide 0 by 5 to find the remainder, which is 0. In other words, since 0 is divisible by 5, the remainder should be 0.
| What is the remainder when 17,475,482 is divided by 5? |
The last digit (unit’s digit) of 17,475,482 is 2. Divide 2 by 5 to find the remainder, which is 2.
Recall that when 2 is divided by 5, you can find the remainder by making the quotient 0.
Remainder Rule when dividing by 8.
To find the remainder when dividing a number by 8, first form a three-digit number from the last three digits of the integer. Then divide the three-digit number by 8 and find the remainder using long division.
For example, take 3,242,353,729. Its last three digits form a three-digit number, 729. Divide 729 by 8 using the long division. You will get a remainder of 1.
| What is the remainder when 17,475,004 is divided by 8? |
The last three digits of 17,475,004 are 004, which is equivalent to 4.
(Recall that you can attach zeros to the beginning of a number without changing its value. So, 004 and 4 are basically the same.)
Then divide 4 by 8. You get a remainder of 4 by making the quotient 0.
Remainder Rule when dividing by 9 – An integer and the sum of its digits both leave the same remainder on division by 9.
Therefore, to find the remainder when an integer is divided by 9, first find the sum of its digits. Then divide the sum by 9 to find the remainder.
For example, take 53. The sum of its digits is 5 + 3 = 8. Now divide the sum (8) by 9 to find the remainder, which is 8. (by making the quotient 0)
Using long division, when we divide 53 by 9, we get the same remainder, which is 8.
This method can be extremely helpful with large numbers.
Let’s take another example – 34,259,677,899.
The sum of the digits is 3+4+2+5+9+6+7+7+8+9+9 = 69. When we divide 69 by 9, we get a remainder of 6.
- Note that you could also have again taken the sum of the digits of 69, 6+9=15, and then divided 15 by 9 to get the same remainder, which is 6.
Note that this rule is quite similar to the remainder rule of 3, but with the condition that the sum of digits should then be divisible by 9.
| What is the remainder when 98,798,978 is divided by 9? |
The sum of the digits of 98,798,978 is:
9+8+7+9+8+9+7+8 = 65. Now, the sum of the digits of 65 is 6+5 =11. When we divide 11 by 9, we get a remainder of 2.
This is the same remainder that you would get if you do the long division on 98,798,978 ÷ 9.
Remainder Rule when dividing by 10
This is the easiest of all the remainder rules – the remainder when an integer is divided by 10 is equal to the last digit (the unit’s digit) of the integer. That’s it.
For example,
The last digit (the unit’s digit) of 489 is 9. Therefore, the remainder when 489 is divided by 10 is equal to 9.
| What is the remainder when -98,798,970 is divided by 10? |
The last digit (the unit’s digit) of -98,798,970 is 0. Therefore, the remainder when -98,798,970 is divided by 10 is equal to 0.
Since -98,798,970 is divisible by 10, therefore the remainder is 0.